Semi-exactness of the equivariant Kasparov bifunctor
Let \(G\) be a locally compact group and \(A,B\) be \(G\)-algebras, i.e., \(C^*\)-algebras such that \(G\) acts continuously (in norm) by automorphisms. In the paper ``Equivariant \(KK\)-theory and the Novikov conjecture'' [Invent. Math. 91, No. 1, 147-201 (1988; Zbl 0647.46053)], \textit{G. G. Kasparov} associates to a pair \((A,B)\) of \(\mathbb{Z}/2\mathbb{Z}\)-graduated \(G\)-algebras the abelian group \(KK_G(A,B)\). The aim of this paper is the study of the behaviour of the bifunctor \(KK_G(\cdot,\cdot)\) with respect to exact sequences. Namely, a notion of strong \(K\)-theoretic amenability for \(G\) is defined and all results obtained concerning the behaviour \(KK_G(\cdot,\cdot)\) with respect to exact sequences are generalized to the case \(KK_G(\cdot,\cdot)\) for \(G\) strongly \(K\)-amenable. Moreover, the nuclearity in Skandalis' sense [see \textit{G. Skandalis}, ``Une notion de nucléarité en \(K\)-théorie (d'après J. Cuntz)'', \(K\)-Theory 1, No. 6, 549-573 (1988; Zbl 0653.46065)] is studied in the equivariant context and also the behaviour of the bifunctor \(KK_G\) with respect to exact sequences is given.
scientific article; zbMATH DE number 1275466